Editing Apps/RotatingPolytropes (section)

=Rotationally Flattened Polytropes=
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Here we review what has been learned over the past century+ regarding the structural properties of rotationally flattened polytropes having indexes in the range, <math>~0 < n < \infty</math>.  Separate chapters are devoted to configurations having the "bookend" index values of <math>~n = 0</math> ([[Apps/MaclaurinSpheroids#Maclaurin_Spheroids_.28axisymmetric_structure.29|incompressible]]) and <math>~n = \infty</math> ([[Apps/HayashiNaritaMiyama82#Rotationally_Flattened_Isothermal_Structures|isothermal]]), and to configurations obeying the [[Apps/OstrikerBodenheimerLyndenBell66|white dwarf equation of state]].  Our review will be divided chronologically into three parts with, generally speaking, the following two publications serving to identify the approximate location of segmentation boundaries:
&nbsp;<br />
&nbsp;<br />
&nbsp;<br />
&nbsp;<br />
&nbsp;<br />

<ul>
<li>[https://ui.adsabs.harvard.edu/abs/1964ApJ...140..552J/abstract R. A. James (1964)], which serves as the first notable example of an astrophysicist drawing upon the capabilities of a digital computer to solve  the set of equations that define &#8212; without approximation &#8212;  the equilibrium structure of axisymmetric, rotationally flattened polytropes.  (Rotating white dwarf structures are also examined.)
</li>
<li>[https://ui.adsabs.harvard.edu/abs/1986ApJS...61..479H/abstract I. Hachisu (1986a)], in which a versatile, efficient, and accurate technique for constructing rotating equilibrium configurations &#8212; hereafter referred to as the ''Hachisu Self-Consistent Field'' ([[AxisymmetricConfigurations/HSCF#Introduction|HSCF]]) technique &#8212;  was detailed.  In retrospect, we recognize that this numerical tool was used by many groups to construct excellent initial equilibrium models for numerical stability studies.
</li>
</ul>  

Because equilibrium models along the incompressible (n = 0) ''Maclaurin'' sequence can be fully described analytically, we should not be surprised to find that Maclaruin spheroids are often used as a base of comparison when the properties of rotationally flattened, ''compressible'' configurations are discussed.  Drawing from the review by [https://ui.adsabs.harvard.edu/abs/1967ARA%26A...5..465L/abstract N. R. Lebovitz (1967)], here are a few relevant features to keep in mind:
<ul>
<li>According to ''Lichtenstein's theorem'', each <font color="green">configuration has a plane of symmetry perpendicular to the axis of rotation; this is the equatorial plane.</font>
</li>
<li>Maclaurin spheroids are a subset of the broader class of equilibrium configurations referred to as Riemann S-type ellipsoids; specifically, the subset containing only axisymmetric configurations.
</li>
<li>Along the Maclaurin equilibrium sequence, each configuration is an oblate spheroid whose position along the sequence can be uniquely specified by the spheroid's axis ratio, b/a, or alternatively by the spheroid's eccentricity, <math>~e \equiv \sqrt{1 - a_3^2/a_1^2}</math>.
</li>
<li>When the angular momentum of the equilibrium configuration, <math>~J_\mathrm{Mac} = \tfrac{2}{5}Ma_1^2 \Omega</math> , <font color="green">is zero, the Maclaurin figure is &hellip; a sphere.  As <math>~J_\mathrm{Mac}</math> increases, the figure flattens progressively, approaching an infinitely thin circular disk as <math>~J_\mathrm{Mac}</math> approaches infinity.</font>
</li>
</ul>

==Prior to the work of James (1964)==

[https://ui.adsabs.harvard.edu/abs/1967ARA%26A...5..465L/abstract N. R. Lebovitz (1967)] offers a concise review of efforts that were made to construct models of rotationally flattened polytropic structures prior to work of [https://ui.adsabs.harvard.edu/abs/1964ApJ...140..552J/abstract R. A. James (1964)].  We will draw heavily from this review.

==Distortions Introduced by Uniform Rotation==
Let's take the following steps &hellip;
* Detail the results from [https://ui.adsabs.harvard.edu/abs/1933MNRAS..93..390C/abstract S. Chandrasekhar (1933)], MNRAS, 93, 390:  ''The equilibrium of distorted polytropes.  I.  The rotational problem''
<table border="0" align="center" width="100%" cellpadding="1"><tr>
<td align="center" width="5%">&nbsp;</td><td align="left">
The purpose of this paper is <font color="green">&hellip; to extend Emden's [work] to the case of rotating gas spheres which in their non-rotating states have polytropic distributions described by the so-called Emden functions.  &hellip; the gas sphere is set rotating at a constant small angular velocity <math>~\omega</math>. &hellip; we shall assume that the rotation is so slow that the configurations are only slightly oblate.</font>
</td></tr></table>
* Compare with the similar analysis for n = 3 conducted earlier by [https://ui.adsabs.harvard.edu/abs/1923MNRAS..83..118M/abstract E. A. Milne (1923)], MNRAS, 83, 118: ''The Equilibrium of a Rotating Star''
* Use as a point of comparison for the more recent work by [https://ui.adsabs.harvard.edu/abs/1985Ap%26SS.113..125C/abstract R. Caimmi (1985)], Astrophysics and Space Science, 113, 125:  ''Emden-Chandrasekhar Axisymmetric, Rigidly Rotating Polytropes.  III. Determination of Equilibrium Configurations by an Improvement of Chandrasekhar's Method''

===Principle Governing Equations===
<!--
We begin with an [[AxisymmetricConfigurations/PGE#Governing_Equations_.28SPH..29|Eulerian formulation of the principle governing equations written in spherical coordinates for an axisymmetric configuration]], namely,

<div align="center">
<span id="Continuity"><font color="#770000">'''Equation of Continuity'''</font></span><br />

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
\frac{\partial \rho}{\partial t}   
+ \biggl[ \frac{1}{r^2} \frac{\partial (\rho r^2 \dot{r})}{\partial r} 
+ \frac{1}{r\sin\theta} \frac{\partial }{\partial\theta} \biggl( \rho \dot\theta r \sin\theta \biggr)
 \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0</math>
  </td>
</tr>
</table>


<span id="PGE:Euler">The Two Relevant Components of the<br />
<font color="#770000">'''Euler Equation'''</font>
</span><br />

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right"><math>~{\hat{e}}_r</math>: &nbsp; &nbsp;</td>
  <td align="right">
<math>
\biggl\{ \frac{\partial \dot{r}}{\partial t} + \biggl[ \dot{r} \frac{\partial \dot{r}}{\partial r} \biggr] + \biggl[ \dot\theta \frac{\partial \dot{r}}{\partial\theta} \biggr] \biggr\}
-  r {\dot\theta}^2
</math>
  </td>
  <td align="center">
=
  </td>
  <td align="left">
<math>
- \biggl[ \frac{1}{\rho} \frac{\partial P}{\partial r}+  \frac{\partial \Phi }{\partial r} \biggr]  + \biggl[ \frac{j^2}{r^3 \sin^2\theta} \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right"><math>~{\hat{e}}_\theta</math>: &nbsp; &nbsp;</td>
  <td align="right">
<math>
r \biggl\{ \frac{\partial \dot{\theta}}{\partial t} + \biggl[ \dot{\theta} \frac{\partial \dot{\theta}}{\partial r} \biggr] + \biggl[ \dot\theta \frac{\partial \dot{r}}{\partial\theta} \biggr] \biggr\} + 2\dot{r} \dot\theta 
</math>
  </td>
  <td align="center">
=
  </td>
  <td align="left">
<math>
- \biggl[ \frac{1}{\rho r}  \frac{\partial P}{\partial\theta} +  \frac{1}{r} \frac{\partial \Phi}{\partial\theta}  \biggr] +  \biggl[ \frac{j^2}{r^3 \sin^3\theta} \biggr] \cos\theta
</math>
  </td>
</tr>
</table>

<span id="PGE:AdiabaticFirstLaw">Adiabatic Form of the<br />
<font color="#770000">'''First Law of Thermodynamics'''</font></span><br />

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
\biggl\{ \frac{\partial \epsilon}{\partial t} + \biggl[ \dot{r} \frac{\partial \epsilon}{\partial r} \biggr] + \biggl[ \dot\theta \frac{\partial \epsilon}{\partial\theta} \biggr]  \biggr\}
+ P\biggl\{ \frac{\partial }{\partial t} \biggl( \frac{1}{\rho}\biggr) 
+ \biggl[ \dot{r} \frac{\partial }{\partial r} \biggl( \frac{1}{\rho}\biggr) \biggr] 
+ \biggl[ \dot\theta \frac{\partial }{\partial\theta} \biggl( \frac{1}{\rho}\biggr) \biggr]  \biggr\}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0</math>
  </td>
</tr>
</table>


<span id="PGE:Poisson"><font color="#770000">'''Poisson Equation'''</font></span><br />

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
\frac{1}{r^2} \frac{\partial }{\partial r} \biggl[ r^2 \frac{\partial \Phi }{\partial r} \biggr] 
+ \frac{1}{r^2 \sin\theta} \frac{\partial }{\partial \theta}\biggl(\sin\theta ~ \frac{\partial \Phi}{\partial\theta}\biggr) 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~4\pi G\rho</math>
  </td>
</tr>
</table>

</div>

where the pair of [[AxisymmetricConfigurations/PGE#RelevantSphericalComponents|"relevant" components of the Euler equation]] have been written in terms of the specific angular momentum,
<div align="center">
<math>~j(r,\theta) \equiv (r\sin\theta)^2 \dot\varphi</math>,
</div>
which is a conserved quantity in axisymmetric systems.  

Given that our aim is to construct steady-state configurations, we should set the partial time-derivative of all scalar quantities to zero; in addition, we will assume that both meridional-plane velocity components, <math>\dot{r}</math> and <math>~\dot{\theta}</math>, to zero &#8212; initially as well as for all time.  As a result of these imposed conditions, both the equation of continuity and the first law of thermodynamics are automatically satisfied; the Poisson equation remains unchanged; and the left-hand-sides of the pair of relevant components of the Euler equation go to zero.  The governing relations then take the following, considerably simplified form:
-->

<table align="center" border="1" cellpadding="10">
<tr>
  <th align="center">Spherical Coordinate Base</th>
</tr>
<tr><td align="center">

<span id="PGE:Poisson"><font color="#770000">'''Poisson Equation'''</font></span><br />

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
\frac{1}{r^2} \frac{\partial }{\partial r} \biggl[ r^2 \frac{\partial \Phi }{\partial r} \biggr] 
+ \frac{1}{r^2 \sin\theta} \frac{\partial }{\partial \theta}\biggl(\sin\theta ~ \frac{\partial \Phi}{\partial\theta}\biggr) 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~4\pi G\rho</math>
  </td>
</tr>
</table>

<span id="PGE:Euler">The Two Relevant Components of the<br />
<font color="#770000">'''Euler Equation'''</font>
</span><br />

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right"><math>~{\hat{e}}_r</math>: &nbsp; &nbsp;</td>
  <td align="right">
<math>
~0
</math>
  </td>
  <td align="center">
=
  </td>
  <td align="left">
<math>
- \biggl[ \frac{1}{\rho} \frac{\partial P}{\partial r}+  \frac{\partial \Phi }{\partial r} \biggr]  + \biggl[ \frac{j^2}{r^3 \sin^2\theta} \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right"><math>~{\hat{e}}_\theta</math>: &nbsp; &nbsp;</td>
  <td align="right">
<math>
~0
</math>
  </td>
  <td align="center">
=
  </td>
  <td align="left">
<math>
- \biggl[ \frac{1}{\rho r}  \frac{\partial P}{\partial\theta} +  \frac{1}{r} \frac{\partial \Phi}{\partial\theta}  \biggr] +  \biggl[ \frac{j^2}{r^3 \sin^3\theta} \biggr] \cos\theta
</math>
  </td>
</tr>
</table>

</td></tr></table>

===Choices and Adaptations===

Among the set of [[AxisymmetricConfigurations/SolutionStrategies#Simple_Rotation_Profile_and_Centrifugal_Potential|simple rotation profiles]] that have been adopted by various research groups over the years, [https://ui.adsabs.harvard.edu/abs/1923MNRAS..83..118M/abstract Milne (1923)], [https://ui.adsabs.harvard.edu/abs/1933MNRAS..93..390C/abstract Chandrasekhar (1933)] and [https://ui.adsabs.harvard.edu/abs/1964ApJ...140..552J/abstract James (1964)] all choose what would generally be considered the simplest, which is the assumption of uniform rotation <math>~(\dot\varphi = \omega_0)</math>.  This means that,

<div align="center">
<math>~j^2 = r^4 \sin^4\theta \omega_0^2 \, .</math>
</div>

And in place of the co-latitude, <math>~\theta</math>, they all adopt the coordinate,
<div align="center">
<math>~\mu \equiv \cos\theta ~~~\Rightarrow ~~~ \frac{\partial}{\partial\mu} = - \frac{\partial}{\sin\theta \partial\theta} \, .</math>
</div>
As a result, the set of three remaining scalar governing equations becomes,

<table border="1" cellpadding="10" align="center" width="40%"><tr><td align="center">
<font color="#770000">'''Poisson Equation'''</font><br />

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
\frac{1}{r^2} \frac{\partial }{\partial r} \biggl[ r^2 \frac{\partial \Phi }{\partial r} \biggr] 
+ \frac{1}{r^2} \frac{\partial }{\partial \mu}\biggl[ (1-\mu^2) \frac{\partial \Phi}{\partial\mu}\biggr] 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~4\pi G\rho</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
[https://ui.adsabs.harvard.edu/abs/1933MNRAS..93..390C/abstract Chandrasekhar (1933)], p. 391, Eq. (4')<br />
[https://ui.adsabs.harvard.edu/abs/1964ApJ...140..552J/abstract James (1964)], p. 553, Eq. (2.1)
  </td>
</tr>
</table>

<font color="#770000">'''The Two Relevant Components of the Euler Equation'''</font>
<br />

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right"><math>~{\hat{e}}_r</math>: &nbsp; &nbsp;</td>
  <td align="right">
<math>
\rho \omega_0^2 r (1-\mu^2)
</math>
  </td>
  <td align="center">
=
  </td>
  <td align="left">
<math>
+ \biggl[ \frac{\partial P}{\partial r} +  \rho \frac{\partial \Phi }{\partial r} \biggr]  
</math>
  </td>
</tr>

<tr>
  <td align="right"><math>~{\hat{e}}_\theta</math>: &nbsp; &nbsp;</td>
  <td align="right">
<math>
\rho \omega_0^2 r^2 \mu
</math>
  </td>
  <td align="center">
=
  </td>
  <td align="left">
<math>
- \biggl[ \frac{\partial P}{\partial\mu} +  \rho\frac{\partial \Phi}{\partial\mu}  \biggr] 
</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="4">
[https://ui.adsabs.harvard.edu/abs/1923MNRAS..83..118M/abstract Milne (1923)], top of p. 126<br />
[https://ui.adsabs.harvard.edu/abs/1933MNRAS..93..390C/abstract Chandrasekhar (1933)], p. 391, Eq. (3')<br />
[https://ui.adsabs.harvard.edu/abs/1964ApJ...140..552J/abstract James (1964)], p. 553, Eqs. (2.2) &amp; (2.3)
  </td>
</tr>
<tr>
  <td align="left" colspan="4">
NOTES:  
* In place of <math>~P</math>, Milne uses <math>~W</math>;
* For the gravitational potential, Milne and Chandrasekhar both adopt the convention, <math>~V \equiv -\Phi </math>, while James adopts the notation, <math>~\Psi \equiv - \Phi</math>;
* James includes terms with azimuthal derivatives in his equation set; these terms are set to zero (as reflected here) when seeking axisymmetric structures.
  </td>
</tr>
</table>
</td></tr></table>


===Chandrasekhar's Approach===

Using the two components of the Euler equation to express spatial derivatives of the gravitational potential in terms of spatial derivatives of the gas pressure, that is, rewriting them in the form,

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right"><math>~{\hat{e}}_r</math>: &nbsp; &nbsp;</td>
  <td align="right">
<math>
\frac{\partial \Phi }{\partial r}
</math>
  </td>
  <td align="center">
=
  </td>
  <td align="left">
<math>
-  \frac{1}{\rho} ~\frac{\partial P}{\partial r} +  \omega_0^2 r (1-\mu^2)  
</math>
  </td>
</tr>

<tr>
  <td align="right"><math>~{\hat{e}}_\theta</math>: &nbsp; &nbsp;</td>
  <td align="right">
<math>
\frac{\partial \Phi}{\partial\mu}  
</math>
  </td>
  <td align="center">
=
  </td>
  <td align="left">
<math>
- \frac{1}{\rho} ~\frac{\partial P}{\partial\mu} - \omega_0^2 r^2 \mu  
</math>
  </td>
</tr>
</table>

we can fold them into the Poisson equation to obtain a 2<sup>nd</sup>-order PDE that relates <math>~\rho(r, \theta)</math> to <math>~P(r, \theta)</math>, namely,

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~4\pi G\rho</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{r^2} \frac{\partial }{\partial r} \biggl\{ r^2 \biggl[-  \frac{1}{\rho} ~\frac{\partial P}{\partial r} +  \omega_0^2 r (1-\mu^2)\biggr]  \biggr\} 
+ \frac{1}{r^2} \frac{\partial }{\partial \mu}\biggl\{ (1-\mu^2) \biggl[ - \frac{1}{\rho} ~\frac{\partial P}{\partial\mu} - \omega_0^2 r^2 \mu  \biggr] \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \frac{1}{r^2} \frac{\partial }{\partial r} \biggl[\frac{r^2}{\rho} ~\frac{\partial P}{\partial r} \biggr]  
+\frac{1}{r^2} \frac{\partial }{\partial r} \biggl[\omega_0^2 r^3 (1-\mu^2)\biggr]  
- \frac{1}{r^2} \frac{\partial }{\partial \mu} \biggl[ \frac{(1-\mu^2)}{\rho} ~\frac{\partial P}{\partial\mu}  \biggr] 
- \frac{1}{r^2} \frac{\partial }{\partial \mu} \biggl[ \omega_0^2 r^2 \mu (1-\mu^2)   \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \frac{1}{r^2} \frac{\partial }{\partial r} \biggl[\frac{r^2}{\rho} ~\frac{\partial P}{\partial r} \biggr]  
- \frac{1}{r^2} \frac{\partial }{\partial \mu} \biggl[ \frac{(1-\mu^2)}{\rho} ~\frac{\partial P}{\partial\mu}  \biggr] 
+3\omega_0^2 (1-\mu^2) - \omega_0^2  (1-3\mu^2) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ 2\omega_0^2 -4\pi G\rho</math>
   </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{r^2} \frac{\partial }{\partial r} \biggl[\frac{r^2}{\rho} ~\frac{\partial P}{\partial r} \biggr]  
+ \frac{1}{r^2} \frac{\partial }{\partial \mu} \biggl[ \frac{(1-\mu^2)}{\rho} ~\frac{\partial P}{\partial\mu}  \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="center" colspan="3">
[https://ui.adsabs.harvard.edu/abs/1933MNRAS..93..390C/abstract Chandrasekhar (1933)], p. 391, Eq. (5)
  </td>
</tr>
</table>

[https://ui.adsabs.harvard.edu/abs/1933MNRAS..93..390C/abstract Chandrasekhar (1933)] refers to this last expression as "the fundamental equation of the problem."  If, following Chandrasekhar's lead, we adopt a [[SR#Barotropic_Structure|polytropic equation of state]],

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~P</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~K\rho^{1+1/n} \, ,</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
[https://ui.adsabs.harvard.edu/abs/1933MNRAS..93..390C/abstract Chandrasekhar (1933)], p. 392, Eq. (6)
  </td>
</tr>
</table>
</div>

this "fundamental equation" can be rewritten strictly in terms of the configuration's axisymmetric density distribution, <math>~\rho(r,\theta)</math>.  Chandrasekhar first adopts a dimensionless function, <math>~\Theta(\xi, \mu)</math>, that is related to the normalized density via the expression,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Theta</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{\rho}{\rho_c}  \biggr)^{1/n}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ P = K \rho^{1+1/n}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\rho_c^{1+1/n} K \Theta^{n+1} \, .
</math>
  </td>
</tr>
</table>
(In doing this, Chandrasekhar is obviously adopting an a path toward solution that parallel's the familiar method used to determine the [[SSC/Structure/Polytropes#Polytropic_Spheres|structure of isolated, nonrotating polytropes]].)  Adopting the additional pair of dimensionless variables,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\xi</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~r \biggl[ \frac{(n+1)K}{4\pi G} ~\rho_c^{1/n - 1} \biggr]^{-1 / 2} </math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; 
  <td align="right">
<math>~v</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{\omega_0^2}{2\pi G \rho_c} </math>
  </td>
</tr>
</table>

Chandrasekhar's "fundamental equation" becomes,
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~
\frac{1}{\xi^2} \frac{\partial }{\partial \xi} \biggl[\xi^2 ~\frac{\partial \Theta}{\partial \xi} \biggr]  
+ \frac{1}{\xi^2} \frac{\partial }{\partial \mu} \biggl[ (1-\mu^2) ~\frac{\partial \Theta}{\partial\mu}  \biggr] 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~v -\Theta^n \, .</math>
   </td>
</tr>
<tr>
  <td align="center" colspan="3">
[https://ui.adsabs.harvard.edu/abs/1933MNRAS..93..390C/abstract Chandrasekhar (1933)], p. 392, Eq. (11)
  </td>
</tr>
</table>

===Milne's Additional Specificity===

====Selected Equation of State====

In his examination of the effects of (uniform) rotation on the equilibrium structure of an otherwise spherically symmetric star, [https://ui.adsabs.harvard.edu/abs/1923MNRAS..83..118M/abstract Milne (1923)] focused on models in which the pressure,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~P_\mathrm{tot}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~P_\mathrm{gas} + P_\mathrm{rad} \, .</math>
  </td>
</tr>
</table>

When a dimensionless parameter, <math>~\beta</math>, is used to quantify the ratio of the gas pressure to the total pressure &#8212; that is, if we set

<div align="center">
<math>~\beta \equiv \frac{P_\mathrm{gas}}{P_\mathrm{tot}} ~~~~\Rightarrow ~~~~ \frac{P_\mathrm{rad}}{P_\mathrm{tot}} = (1-\beta) \, ,</math>
</div>

then, as we have [[SSC/Structure/BiPolytropes/Analytic1.5_3#Envelope|detailed in our separate discussion]] of [http://adsabs.harvard.edu/abs/1930MNRAS..91....4M Milne's (1930)] early work on bipolytropic stellar models, the pressure-density relation and the temperature-density relation become, respectively,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~P</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\biggl[ \biggl( \frac{\Re}{\mu_e}\biggr)^4 \biggl(\frac{1-\beta}{\beta^4}\biggr) \frac{3}{a_\mathrm{rad}} \biggr]^{1/3} \rho^{1 + 1/3}</math>
  </td>
  <td align="center">&nbsp; &nbsp; and, &nbsp; &nbsp;</td>
  <td align="right">
<math>~\frac{T^3}{\rho}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>\biggl[ \biggl( \frac{\Re}{\mu_e}\biggr) \biggl(\frac{1-\beta}{\beta}\biggr) \frac{3}{a_\mathrm{rad}} \biggr] \, .</math>
  </td>
</tr>
</table>

Now, when building a realistic stellar model, one must expect that, in general, the parameter <math>~\beta</math> will vary with position throughout the model.  But ''if'' the assumption is made that <math>~\beta</math> has the same value throughout the equilibrium configuration, then  we are effectively adopting a polytropic equation of state with,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~n</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~3 \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~K</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \biggl( \frac{\Re}{\mu_e}\biggr)^4 \biggl(\frac{1-\beta}{\beta^4}\biggr) \frac{3}{a_\mathrm{rad}} \biggr]^{1/3} \, .</math>
  </td>
</tr>
</table>

With this realization, [https://ui.adsabs.harvard.edu/abs/1933MNRAS..93..390C/abstract Chandrasekhar's (1933)] work should be considered a ''generalization'' of [https://ui.adsabs.harvard.edu/abs/1923MNRAS..83..118M/abstract Milne's (1923)] modeling effort.  Also, the results of the latter's work should match Chandrasekhar's results for the specific case of a rotating <math>~n=3</math> polytrope.

====Radiative Equilibrium====

[https://ui.adsabs.harvard.edu/abs/1923MNRAS..83..118M/abstract Milne (1923)] made an effort to ensure that his equilibrium models were not only in hydrostatic balance but that they also were in "radiative equilibrium;"  see especially his &sect;II.9.  Drawing on ''[[PGE/FirstLawOfThermodynamics#Example_B|Example B]]'' from our introductory discussion of [[PGE/FirstLawOfThermodynamics#Nonadiabatic_Environments|nonadiabatic environments]], Milne accomplished this by, effectively, adopting the steady-state specific-entropy expression,

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\rho T \cancelto{0}{\frac{ds_\mathrm{tot}}{dt}} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\rho \epsilon_\mathrm{nuc} - \nabla \cdot \vec{F}_\mathrm{rad} \, .
</math>
  </td>
</tr>
</table>

In this expression, <math>~\epsilon_\mathrm{nuc}(\rho,T)</math> specifies the rate at which (specific) energy is released via thermonuclear reactions, and

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\vec{F}_\mathrm{rad}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{c}{3\rho\kappa_R} \nabla (a_\mathrm{rad}T^4) \, .</math>
  </td>
</tr>
</table>

Given that the energy per unit volume in the radiation field is, <math>~E_\mathrm{rad} = a_\mathrm{rad} T^4</math>, this "radiative equilibrium" condition may be rewritten as,

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
\nabla \cdot \biggl[ \frac{1}{3\rho\kappa_R} \nabla E_\mathrm{rad} \biggr] 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{\rho \epsilon_\mathrm{nuc} }{c} \, .</math>
  </td>
</tr>
</table>
This is identical to equation (7) of [https://ui.adsabs.harvard.edu/abs/1923MNRAS..83..118M/abstract Milne (1923)] except:  (a) His divergence and gradient operators appear as they would in Cartesian coordinates; and (b) an extra factor of <math>~4\pi</math> appears in the term on the right-hand side of his expression.  We attribute the extra factor of <math>~4\pi</math> to slightly different definitions of the energy derived from nuclear reactions; specifically, we suspect that, <math>~\epsilon_\mathrm{nuc} = 4\pi \epsilon_\mathrm{Milne}</math>, because immediately following his equation (5) &#8212; near the top of his p. 123 &#8212; we find the sentence &hellip; "<font color="green">Now suppose that <math>~4\pi\epsilon</math> is the energy evolved per unit mass per second at</font> [a given location]."

====Compare Radiative Equilibrium with Mechanical Equilibrium====

Given that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\tfrac{1}{3} E_\mathrm{rad}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~P_\mathrm{rad} = (1-\beta) P_\mathrm{tot} \, ,</math>
  </td>
</tr>
</table>
the radiative equilibrium condition can be rewritten as,

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
\nabla \cdot \biggl\{ \frac{1}{\rho\kappa_R} \nabla \biggl[(1-\beta)P_\mathrm{tot}\biggr] \biggr\} 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{\rho \epsilon_\mathrm{nuc} }{c} \, .</math>
  </td>
</tr>
</table>

If we now express the differential operators in terms of [[AxisymmetricConfigurations/PGE#Spherical_Coordinate_Base|spherical coordinates]] and (for the time being) assume that <math>~\kappa_R</math> and <math>~\beta</math> are both independent of position, this becomes,

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~- \frac{\rho \epsilon_\mathrm{nuc}\kappa_R }{c~(1-\beta)} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\nabla \cdot \biggl\{  \biggl[{\hat{e}}_r \biggl[ \frac{1}{\rho } \frac{\partial P_\mathrm{tot}}{\partial r}\biggr] + {\hat{e}}_\theta \biggl[\frac{1}{\rho r} \frac{\partial P_\mathrm{tot}}{\partial \theta}\biggr] \biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{r^2}\frac{\partial}{\partial r}\biggl[ \frac{r^2}{\rho } \frac{\partial P_\mathrm{tot}}{\partial r}\biggr] 
+ \frac{1}{r\sin\theta} \frac{\partial}{\partial\theta}\biggl[\frac{\sin\theta}{\rho r} \frac{\partial P_\mathrm{tot}}{\partial \theta}\biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{r^2}\frac{\partial}{\partial r}\biggl[ \frac{r^2}{\rho } \frac{\partial P_\mathrm{tot}}{\partial r}\biggr] 
+ \frac{1}{r} \frac{\partial}{\partial\mu}\biggl[\frac{(1-\mu^2)}{\rho r} \frac{\partial P_\mathrm{tot}}{\partial \mu}\biggr] \, .
</math>
  </td>
</tr>
</table>

Next, following [https://ui.adsabs.harvard.edu/abs/1923MNRAS..83..118M/abstract Milne's (1923)] lead, let's assume that the pressure, <math>~P</math>, that appears in Chandrasekhar's "fundamental equation" is only slightly different from <math>~P_\mathrm{tot}</math> &#8212; that is, let's write,

<div align="center">
<math>~P = P_\mathrm{tot} + \delta P \, ,</math>
</div>

then subtract the derived radiative equilibrium relation from Chandrasekhar's "fundamental equation."  Doing this, we obtain,

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~2\omega_0^2 -4\pi G\rho + \frac{\rho \epsilon_\mathrm{nuc}\kappa_R }{c~(1-\beta)} </math>
   </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{r^2} \frac{\partial }{\partial r} \biggl[\frac{r^2}{\rho} ~\frac{\partial (\delta P)}{\partial r} \biggr]  
+ \frac{1}{r^2} \frac{\partial }{\partial \mu} \biggl[ \frac{(1-\mu^2)}{\rho} ~\frac{\partial (\delta P)}{\partial\mu}  \biggr] \, .
</math>
  </td>
</tr>
</table>
Notice that, if we specifically choose the value of <math>~\beta</math> such that (see [https://ui.adsabs.harvard.edu/abs/1923MNRAS..83..118M/abstract Milne's] &sect;I.6),
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\beta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1 - \frac{\kappa_R (\epsilon_\mathrm{nuc}/4\pi)}{c~G} \, ,</math>
  </td>
</tr>
</table>
then the left-hand side of this relation simplifies considerably.  Specifically, we end up with,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~2</math>
   </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{r^2} \frac{\partial }{\partial r} \biggl[\frac{r^2}{\rho} ~\frac{\partial U}{\partial r} \biggr]  
+ \frac{1}{r^2} \frac{\partial }{\partial \mu} \biggl[ \frac{(1-\mu^2)}{\rho} ~\frac{\partial U}{\partial\mu}  \biggr] \, ,
</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
[https://ui.adsabs.harvard.edu/abs/1923MNRAS..83..118M/abstract Milne (1923)], p. 125, Eq. (20)
  </td>
</tr>
</table>
where we have adopted the variable notation used by [https://ui.adsabs.harvard.edu/abs/1923MNRAS..83..118M/abstract Milne (1923)], viz., <math>~\delta P \rightarrow \omega_0^2 U \, .</math>  And, simultaneously,  the condition for radiative equilibrium takes the form,

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~- \frac{\rho \epsilon_\mathrm{nuc}\kappa_R }{c} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{r^2}\frac{\partial}{\partial r}\biggl[ \frac{r^2}{3\rho } \frac{\partial E_\mathrm{rad}}{\partial r}\biggr] 
+ \frac{1}{r} \frac{\partial}{\partial\mu}\biggl[\frac{(1-\mu^2)}{3 \rho r} \frac{\partial E_\mathrm{rad}}{\partial \mu}\biggr] 
</math>
  </td>
  <td align="center">&nbsp;</td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ - 4\pi G \rho (1-\beta) </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{r^2}\frac{\partial}{\partial r}\biggl[ \frac{r^2}{3\rho } \frac{\partial (a_\mathrm{rad} T^4)}{\partial r}\biggr] 
+ \frac{1}{r} \frac{\partial}{\partial\mu}\biggl[\frac{(1-\mu^2)}{3 \rho r} \frac{\partial (a_\mathrm{rad} T^4)}{\partial \mu}\biggr] 
</math>
  </td>
  <td align="center">&nbsp;</td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ - \frac{3\pi G \rho (1-\beta)}{a_\mathrm{rad} } </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{r^2}\frac{\partial}{\partial r}\biggl[ \frac{r^2 T^3}{\rho } \frac{\partial T}{\partial r}\biggr] 
+ \frac{1}{r^2} \frac{\partial}{\partial\mu}\biggl[\frac{(1-\mu^2)T^3}{ \rho} \frac{\partial T}{\partial \mu}\biggr] \, .
</math>
  </td>
  <td align="right">
[[File:CommentButton02.png|right|100px|Comment by J. E. Tohline on 25 July 2019:  Milne's Equation (19) does not include the factor of a_rad (the radiation constant) that is shown here in the denominator of the left-hand-side; this appears to be a typesetting error, as Milne's expression is even dimensionally incorrect without this factor.]]
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
[https://ui.adsabs.harvard.edu/abs/1923MNRAS..83..118M/abstract Milne (1923)], p. 125, Eq. (19)
  </td>
  <td align="center">&nbsp;</td>
</tr>
</table>

===Solutions===

In determining the equilibrium configuration's axisymmetric density distribution, <math>~\rho(r,\theta)</math>, Chandrasekhar adopts a dimensionless function, <math>~\Theta(\xi, \mu)</math>, that is related to the normalized density via the expression,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Theta</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{\rho}{\rho_c}  \biggr)^{1/n}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ P = K \rho^{1+1/n}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\rho_c^{1+1/n} K \Theta^{n+1} \, .
</math>
  </td>
</tr>
</table>
(In doing this, Chandrasekhar is obviously adopting an a path toward solution that parallel's the familiar method used to determine the [[SSC/Structure/Polytropes#Polytropic_Spheres|structure of isolated, nonrotating polytropes]].)  He then argues that, for slowly rotating configurations, the function, <math>~\Theta(\xi,\mu)</math> can be effectively expressed as a small perturbation (in the two-dimensional meridional plane), <math>~v\Psi(\xi,\mu) \ll 1</math>, added to the radially dependent "Emden's function," <math>~\theta(\xi)</math>, that defines the structure of non-rotating polytropic configurations; that is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Theta(\xi,\mu)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\theta(\xi) + v \Psi(\xi,\mu) + \mathcal{O}(v^2) \, ,
</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
[https://ui.adsabs.harvard.edu/abs/1933MNRAS..93..390C/abstract Chandrasekhar (1933)], p. 392, Eq. (13)
  </td>
</tr>
</table>

where,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~v</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{\omega_0^2}{2\pi G\rho_c} \ll 1 \, .</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
[https://ui.adsabs.harvard.edu/abs/1933MNRAS..93..390C/abstract Chandrasekhar (1933)], p. 392, Eq. (10)
  </td>
</tr>
</table>

He deduces that, to lowest order in a [https://en.wikipedia.org/wiki/Legendre_polynomials#Rodrigues'_formula_and_other_explicit_formulas Legendre polynomial series],
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Psi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\psi_0(\xi) - \frac{5}{6} ~ \frac{\xi_1^2}{3\psi_2(\xi_1) + \xi_1 \psi_2^'(\xi_1)} \psi_2(\xi) P_2(\mu) \, ,
</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
[https://ui.adsabs.harvard.edu/abs/1933MNRAS..93..390C/abstract Chandrasekhar (1933)], p. 395, Eq. (36)
  </td>
</tr>
</table>
where:  <math>~P_2(\mu) = \tfrac{1}{2}(3\mu^2 - 1)</math>; <math>~\xi_1</math> is "<font color="darkgreen">the first zero of the Emden's function with index <math>~n</math></font>"; and <math>~\psi_0</math> and <math>~\psi_2</math> satisfy the 2<sup>nd</sup>-order ODEs,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{1}{\xi^2} \frac{d}{d\xi} \biggl( \xi^2 \frac{d\psi_0}{d\xi} \biggr)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- n\theta^{n-1} \psi_0 + 1 \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{1}{\xi^2} \frac{d}{d\xi} \biggl( \xi^2 \frac{d\psi_2}{d\xi} \biggr)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( - n\theta^{n-1} + \frac{6}{\xi^2} \biggr) \psi_2 \, .</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
[https://ui.adsabs.harvard.edu/abs/1933MNRAS..93..390C/abstract Chandrasekhar (1933)], p. 395, Eqs. (37<sub>1</sub> &amp; 37<sub>2</sub>)
  </td>
</tr>
</table>

Realizing that "<font color="darkgreen">the boundary <math>~\xi_0</math> is given by <math>~\Theta = 0</math></font>," Chandrasekhar deduced as well that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\xi_0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\xi_1 + \frac{v}{|\theta_1^'|} \biggl[
\psi_0(\xi_1) - \frac{5}{6} ~ \frac{\xi_1^2 \psi_2(\xi_1) P_2(\mu)}{3\psi_2(\xi_1) + \xi_1 \psi_2^'(\xi_1)}  \biggr] \, ,
</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
[https://ui.adsabs.harvard.edu/abs/1933MNRAS..93..390C/abstract Chandrasekhar (1933)], p. 395, Eq. (38)
  </td>
</tr>
</table>